Math Problems Vault

Problem 5003

If $u_{\nu } = v_{\nu +1} - v_{\nu }$ , then we have the identity:

\sum _{\nu =0}^{n-1} u_{\nu } = v_{n} - v_{0}.

This remark and similar related techniques give the solution to the following problems. Prove the identity:

\sum _{\nu =1}^{n} \nu ^{2} = \frac{n(n+1)(2n+1)}{6}.

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Problem 5004

Prove the identity $\sum_{\nu =1}^{n} \nu^{3} = \left(\frac{n(n+1)}{2}\right)^{2}$ .

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Problem 5005

Prove the identity $1^{2}+3^{2}+5^{2}+\ldots +(2n-1)^{2} = \frac{4n^{3}-n}{3}$ .

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Problem 5006

Prove the identity $1^{3}+3^{3}+5^{3}+\ldots +(2n-1)^{3} = 2n^{4}-n^{2}$ .

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Problem 5007

Prove the identity $\sum_{\nu =1}^{n-1} \nu^{2}(n-\nu )^{2} = \frac{n^{5}-n}{30}$ .

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Problem 5008

Prove the identity $1^{4}-2^{4}+3^{4}-\ldots +(-1)^{n-1}n^{4} = (-1)^{n-1}\frac{n^{4}+2n^{3}-n}{2}$ .

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Problem 5009

Prove the identity $s_{5}+s_{7} = 2s_{3}^{2}$ ; $s_{\nu } = 1^{\nu }+2^{\nu }+\ldots +n^{\nu }$ .

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Problem 5010

Using Euler's formulas, find the sum: $1+\cos x+\cos 2x+\ldots +\cos (n-1)x$ .

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Problem 5011

Using Euler's formulas, find the sum: $\cos x+\cos 3x+\ldots +\cos (2n-1)x$ .

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Problem 5012

Using Euler's formulas, find the sum: $\sin x+\sin 2x+\ldots +\sin nx$ .

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Problem 5013

Using Euler's formulas, find the sum: $\sin x-\sin 2x+\ldots -\sin 2nx$ .

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Problem 5014

Using Euler's formulas, find the sum: $\sin^{2}x-\sin^{2}3x+\ldots +(-1)^{n-1}\sin^{2}(2n-1)x$ .

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Problem 5015

Using Euler's formulas, find the sum: $x\sin a+x^{2}\sin 2a+\ldots +x^{n-1}\sin (n-1)a$ .

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Problem 5016

Prove the equality $x+2x^{2}+3x^{3}+4x^{4}+\ldots = \frac{x}{(1-x)^{2}}$ ; $\left|x\right| < 1$ .

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Problem 5017

Prove the equality $1-4x+9x^{2}-16x^{3}+\ldots = \frac{1-x}{(1+x)^{3}}$ ; $\left|x\right| < 1$ .

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Problem 5018

Prove the identity $\frac{1}{a(a+1)}+\frac{1}{(a+1)(a+2)}+\ldots +\frac{1}{(a+n-1)(a+n)} = \frac{1}{a}-\frac{1}{a+n}$ .

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Problem 5019

Prove the identity

\frac{1}{a(a+1)(a+2)}+\frac{1}{(a+1)(a+2)(a+3)}+\ldots +\frac{1}{(a+n-1)(a+n)(a+n+1)} =

= \frac{1}{2a(a+1)} - \frac{1}{2(a+n)(a+n-1)}.

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Problem 5020

Prove the identity

\frac{1}{a(a+p)}+\frac{1}{(a+1)(a+p+1)}+\ldots +\frac{1}{(a+n-1)(a+p+n-1)} =

= \frac{1}{p}\left(\frac{1}{a}+\frac{1}{a+1}+\ldots +\frac{1}{a+p-1}\right) - \frac{1}{p}\left(\frac{1}{a+n}+\frac{1}{a+n+1}+\ldots +\frac{1}{a+n+p-1}\right).

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Problem 5021

Prove the equality $\frac{1}{1\cdot 3}+\frac{1}{3\cdot 5}+\frac{1}{5\cdot 7}+\ldots = \frac{1}{2}$ .

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Problem 5022

Prove the equality $\frac{1}{1\cdot 4}+\frac{1}{4\cdot 7}+\frac{1}{7\cdot 10}+\ldots = \frac{1}{3}$ .

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Problem 5023

Prove the equality $\frac{1}{1\cdot 4}+\frac{1}{2\cdot 5}+\frac{1}{3\cdot 6}+\ldots = \frac{11}{18}$ .

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Problem 5024

Prove the equality $\frac{1}{1\cdot 7}+\frac{1}{3\cdot 9}+\frac{1}{5\cdot 11}+\ldots = \frac{23}{90}$ .

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Problem 5025

Prove the equality $\sum_{n=1}^{\infty } \frac{n}{(2n-1)^{2}(2n+1)^{2}} = \frac{1}{8}$ .

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Problem 5026

Prove the equality $\sum_{n=1}^{\infty } \frac{2n+1}{n^{2}(n+1)^{2}} = 1$ .

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Problem 5027

Prove the equality $\sum_{n=1}^{\infty } \frac{1}{n(n+1)(n+2)} = \frac{1}{4}$ .

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Problem 5028

Prove the equality $\frac{1}{1\cdot 3\cdot 5\cdot 7}+\frac{1}{3\cdot 5\cdot 7\cdot 9}+\frac{1}{5\cdot 7\cdot 9\cdot 11}+\ldots = \frac{1}{90}$ .

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Problem 5029

Prove the equality $\sum_{n=1}^{\infty } \frac{n^{2}}{(n+1)(n+2)(n+3)(n+4)} = \frac{5}{36}$ .

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Problem 5030

Prove the equality $\sum_{n=1}^{\infty } \frac{1}{n(4n^{2}-1)} = 2\ln 2 - 1$ .

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Problem 5031

Prove the equality $\sum_{n=1}^{\infty } \frac{n}{(4n^{2}-1)^{2}} = \frac{1}{8}$ .

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Problem 5032

Using the identity

\operatorname {arctg}\frac{b}{a+n-1} - \operatorname {arctg}\frac{b}{a+n} = \operatorname {arctg}\frac{b}{b^{2}+(a+n)(a+n-1)},

prove the equality $\sum_{n=1}^{\infty } \operatorname {arctg}\frac{1}{2n^{2}} = \frac{\pi }{4}$ .

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Problem 5033

Using the identity

\operatorname {arctg}\frac{b}{a+n-1} - \operatorname {arctg}\frac{b}{a+n} = \operatorname {arctg}\frac{b}{b^{2}+(a+n)(a+n-1)},

prove the equality $\sum_{n=1}^{\infty } \operatorname {arctg}\frac{1}{n^{2}+n+1} = \frac{\pi }{4}$ .

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Problem 5034

Prove that for $\left|x\right| < 1$ we have the equality $(1+x)(1+x^{2})(1+x^{4})(1+x^{8})\ldots = \frac{1}{1-x}$ .

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Problem 5035

Prove that for $\left|x\right| < 1$ we have the equality $\frac{x}{1+x}+\frac{2x^{2}}{1+x^{2}}+\frac{4x^{4}}{1+x^{4}}+\ldots = \frac{x}{1-x}$ .

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Problem 5036

Prove that for $\left|x\right| < 1$ we have the equality $\frac{x}{1-x^{2}}+\frac{x^{2}}{1-x^{4}}+\frac{x^{4}}{1-x^{8}}+\ldots = \frac{x}{1-x}$ .

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Problem 5037

Prove the equality $\frac{1}{2}\operatorname {tg}\frac{x}{2}+\frac{1}{4}\operatorname {tg}\frac{x}{4}+\frac{1}{8}\operatorname {tg}\frac{x}{8}+\ldots = \frac{1}{x}-\operatorname {ctg}x$ .

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Problem 5038

Prove the equality $\sum_{n=1}^{\infty } \frac{1}{2^{2n}\cos^{2}\frac{x}{2^{n}}} = \frac{1}{\sin^{2}x} - \frac{1}{x^{2}}$ .

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Problem 5039

Using the equalities:

\int _{0}^{1} t^{n-1}\, dt = \frac{1}{n}, \quad \int _{0}^{1} t^{n-1}(1-t)\, dt = \frac{1}{n(n+1)}, \quad \int _{0}^{1} t^{n-1}(1-t)^{2}\, dt = \frac{1}{n(n+1)(n+2)},

prove that:

1 - \frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \ldots = \frac{1}{3}\left(\frac{\pi }{\sqrt{3}}+\ln 2\right).

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Problem 5040

Using the equalities:

\int _{0}^{1} t^{n-1}\, dt = \frac{1}{n}, \quad \int _{0}^{1} t^{n-1}(1-t)\, dt = \frac{1}{n(n+1)}, \quad \int _{0}^{1} t^{n-1}(1-t)^{2}\, dt = \frac{1}{n(n+1)(n+2)},

prove that:

1 - \frac{1}{5} + \frac{1}{9} - \frac{1}{13} + \ldots = \frac{1}{4\sqrt{2}}\left(\pi + 2\ln (\sqrt{2}+1)\right).

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Problem 5041

Prove the equality $1+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{1}{7}-\frac{1}{8}+\ldots = \frac{\pi }{4}+\frac{1}{2}\ln 2$ .

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Problem 5042

Prove the equality $1+\frac{1}{3}+\frac{1}{5}-\frac{1}{7}-\frac{1}{9}-\frac{1}{11}+\frac{1}{13}+\ldots = \frac{5\pi }{12}$ .

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Problem 5043

Prove the equality $\sum_{n=1,3,5,\ldots } \frac{(-1)^{\frac{n^{2}-1}{8}}}{n} = \frac{\ln (1+\sqrt{2})}{\sqrt{2}}$ .

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Problem 5044

Prove the equality $\sum_{n=1}^{\infty } \frac{1}{(n+\alpha )(n+\beta )} = \frac{1}{\beta -\alpha }\int_{0}^{1} \frac{x^{\alpha }-x^{\beta }}{1-x}\, dx$ ; $\alpha > -1$ , $\beta > -1$ , $\alpha \neq \beta$ .

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Problem 5045

Prove the equality $\sum_{n=0}^{\infty } \frac{1}{(4n+1)(4n+3)} = \frac{\pi }{8}$ .

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Problem 5046

Prove the equality $\sum_{n=1}^{\infty } \frac{1}{n(n+m)} = \frac{1}{m}\left(1+\frac{1}{2}+\frac{1}{3}+\ldots +\frac{1}{m}\right)$ .

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Problem 5047

Prove the equality $\frac{1}{1\cdot 2\cdot 3}+\frac{1}{3\cdot 4\cdot 5}+\frac{1}{5\cdot 6\cdot 7}+\ldots = \ln 2 - \frac{1}{2}$ .

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Problem 5048

Prove the equality $\frac{1}{1\cdot 2\cdot 3}-\frac{1}{3\cdot 4\cdot 5}+\frac{1}{5\cdot 6\cdot 7}-\ldots = \frac{1}{2}(1-\ln 2)$ .

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Problem 5049

Prove the equality $\frac{1}{1\cdot 2\cdot 3\cdot 4}+\frac{1}{4\cdot 5\cdot 6\cdot 7}+\frac{1}{7\cdot 8\cdot 9\cdot 10}+\ldots = \frac{1}{6}-\frac{1}{4}\ln 3+\frac{\pi }{12\sqrt{3}}$ .

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Problem 5050

Prove the equality $\frac{1}{1^{2}\cdot 3^{2}}+\frac{1}{3^{2}\cdot 5^{2}}+\frac{1}{5^{2}\cdot 7^{2}}+\ldots = \frac{\pi^{2}}{16}-\frac{1}{2}$ .

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Problem 5051

Prove the equality $\frac{1}{1^{2}\cdot 3^{2}\cdot 5^{2}}+\frac{1}{3^{2}\cdot 5^{2}\cdot 7^{2}}+\frac{1}{7^{2}\cdot 9^{2}\cdot 11^{2}}+\ldots = \frac{3\pi^{2}}{256}-\frac{1}{9}$ .

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Problem 5052

Prove the equality $\frac{1}{1^{2}\cdot 2^{2}\cdot 3^{2}}+\frac{1}{2^{2}\cdot 3^{2}\cdot 4^{2}}+\frac{1}{3^{2}\cdot 4^{2}\cdot 5^{2}}+\ldots = \frac{4\pi^{2}-39}{16}$ .

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Problem 5053

Prove the equality $\sum_{n=1}^{\infty } \frac{x^{n}}{n(n+1)(n+2)} = \frac{3}{4}-\frac{1}{2x}-\frac{(1-x)^{2}}{2x^{2}}\ln \frac{1}{1-x}$ .

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Problem 5054

Prove the equality

\sum _{n=0}^{\infty } \frac{x^{n}}{(2n+1)(2n+3)(2n+5)} = \frac{5}{24x}-\frac{1}{8x^{2}}+\frac{(1-x)^{2}}{8x^{2}} \cdot R;

R = \frac{1}{2\sqrt{x}}\ln \frac{1+\sqrt{x}}{1-\sqrt{x}}, \text{ if } x > 0; \quad R = \frac{1}{\sqrt{-x}}\operatorname {arctg}\sqrt{-x}, \text{ if } x < 0.

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Problem 5055

Using Newton's binomial theorem, prove that

1 - \frac{1}{2} + \frac{1\cdot 3}{2\cdot 4} - \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6} + \ldots = \frac{1}{\sqrt{2}}.

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Problem 5056

Using the equality $\int_{0}^{1} t^{n-1}\, dt = \frac{1}{n}$ , obtain that

\frac{2\cdot 4\cdot 6\ldots 2n}{3\cdot 5\cdot 7\ldots (2n+1)} = \int _{0}^{1} (1-t^{2})^{n}\, dt,

and derive the formula:

\frac{2}{3}+\frac{2\cdot 4}{3\cdot 5}x+\frac{2\cdot 4\cdot 6}{3\cdot 5\cdot 7}x^{2}+\ldots = \int _{0}^{1} \frac{1-t^{2}}{1-x+xt^{2}}\, dt; \quad \left|x\right| < 1.

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Problem 5057

Prove the equality $\frac{2}{3}-\frac{2\cdot 4}{3\cdot 5}+\frac{2\cdot 4\cdot 6}{3\cdot 5\cdot 7}-\ldots = 1-\frac{1}{\sqrt{2}}\ln (1+\sqrt{2})$ .

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Problem 5058

Prove the equality $\frac{x^{2}}{3}+\frac{2x^{4}}{3\cdot 5}+\frac{2\cdot 4}{3\cdot 5\cdot 7}x^{6}+\ldots = 1-\frac{\sqrt{1-x^{2}}\arcsin x}{x}$ ; $\left|x\right| < 1$ .

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Problem 5059

Prove the equality $\frac{1}{4}-\frac{1\cdot 3}{4\cdot 6}+\frac{1\cdot 3\cdot 5}{4\cdot 6\cdot 8}-\ldots = 3-2\sqrt{2}$ .

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Problem 5060

Prove the equality $\frac{1}{2\cdot 4}+\frac{1\cdot 3}{2\cdot 4\cdot 6}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 8}+\ldots = \frac{1}{2}$ .

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Problem 5061

Noting that $\int_{0}^{1} t^{n-1}\ln t\, dt = -\frac{1}{n^{2}}$ ; $\int_{0}^{1} t^{n-1}\ln^{2}t\, dt = \frac{2}{n^{3}}$ ; $n > 0$ , prove the equality

1+\frac{1}{2}\cdot \frac{1}{3^{2}}+\frac{1\cdot 3}{2\cdot 4}\cdot \frac{1}{5^{2}}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\cdot \frac{1}{7^{2}}+\ldots = \frac{\pi }{2}\ln 2.

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Problem 5062

Noting that $\int_{0}^{1} t^{n-1}\ln t\, dt = -\frac{1}{n^{2}}$ ; $\int_{0}^{1} t^{n-1}\ln^{2}t\, dt = \frac{2}{n^{3}}$ ; $n > 0$ , prove the equality

1+\frac{1}{2}\cdot \frac{1}{3^{3}}+\frac{1\cdot 3}{2\cdot 4}\cdot \frac{1}{5^{3}}+\ldots = \frac{\pi ^{3}}{48}+\frac{\pi }{4}(\ln 2)^{2}.

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Problem 5063

Starting from the equality $\frac{n!}{a(a+1)\ldots (a+n)} = \int_{0}^{1} (1-t)^{a-1}t^{n}\, dt$ , prove that

\frac{1}{a}+\frac{x}{a(a+1)}+\frac{x^{2}}{a(a+1)(a+2)}+\ldots = \int _{0}^{1} (1-t)^{a-1}e^{xt}\, dt; \quad a > 0.

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Problem 5064

Prove that

\frac{1}{a}+\frac{1}{a(a+1)}+\frac{1}{a(a+1)(a+2)}+\ldots = e\left(\frac{1}{a}-\frac{1}{1!(a+1)}+\frac{1}{2!(a+2)}-\ldots \right).

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Problem 5065

Using the identity

\frac{n!}{x(x+1)\ldots (x+n-1)} = \frac{n!}{(x-2)x(x+1)\ldots (x+n-2)} - \frac{(n+1)!}{(x-2)x(x+1)\ldots (x+n-1)},

prove the equality:

\frac{1}{x}+\frac{2!}{x(x+1)}+\frac{3!}{x(x+1)(x+2)}+\ldots = \frac{1}{x+2}; \quad x > 2.

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Problem 5066

Prove Stirling's equality:

\frac{1}{x(x+1)}+\frac{1+n}{x(x+1)(x+2)}+\frac{(1+n)(2+n)}{x(x+1)(x+2)(x+3)}+\ldots =

= \frac{1}{x^{2}-nx}; \quad x > n.

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Problem 5067

Prove that

\sum _{n=0}^{\infty } \frac{1}{(t+n)^{2}} = \frac{1}{t}+\frac{1}{2t(t+1)}+\frac{2!}{3t(t+1)(t+2)}+\ldots

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Problem 5068

Prove that for $\gamma > \alpha > 0$ , $\left|x\right| < 1$ we have the equality

F(\alpha , \beta , \gamma ; x) = \frac{\Gamma (\gamma )}{\Gamma (\alpha )\Gamma (\gamma -\alpha )} \int _{0}^{1} u^{\alpha -1}(1-u)^{\gamma -\alpha -1}(1-xu)^{-\beta }\, du,

where

F(\alpha , \beta , \gamma ; x) = 1+\frac{\alpha \cdot \beta }{1\cdot \gamma }x+\frac{\alpha (\alpha +1)}{1\cdot 2}\frac{\beta (\beta +1)}{\gamma (\gamma +1)}x^{2}+\ldots

(Gauss's hypergeometric series).

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Problem 5069

Prove that for $\operatorname {Re}(\gamma -\alpha -\beta ) > 0$ we have the equality

F(\alpha , \beta , \gamma ; 1) = \frac{\Gamma (\gamma )\Gamma (\gamma -\alpha -\beta )}{\Gamma (\gamma -\alpha )\Gamma (\gamma -\beta )}

( $\operatorname {Re}$ denotes the real part of a complex number).

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Problem 5070

Prove the equality $1+\frac{a}{b}x+\frac{a(a+1)}{b(b+1)}x^{2}+\ldots = 1+ax\int_{0}^{1} \frac{(1-u)^{b-1}}{(1-ux)^{a-1}}\, du$ ; $b > 0$ .

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Problem 5071

Prove the equality $\frac{1}{2}+\left(\frac{1}{2}\right)^{2}\cdot \frac{1}{4}+\left(\frac{1\cdot 3}{2\cdot 4}\right)^{2}\cdot \frac{1}{6}+\ldots = \frac{2}{\pi }$ .

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Problem 5072

Prove the equality $\left(\frac{1}{2}\right)^{2}+3\cdot \left(\frac{1}{2\cdot 4}\right)^{2}+5\cdot \left(\frac{1\cdot 3}{2\cdot 4\cdot 6}\right)^{2}+\ldots = 1-\frac{2}{\pi }$ .

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Problem 5073

Prove the equality $\varphi (x)+\varphi \left(x+\frac{\pi }{2}\right) = 4\varphi (2x)$ , where

\varphi (x) = \sum _{n=-\infty }^{\infty } \frac{1}{(x-n\pi )^{2}} - \frac{1}{\sin ^{2}x}.

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Problem 5074

Using the result of problem 5073 (the equality $\varphi (x)+\varphi \left(x+\frac{\pi }{2}\right) = 4\varphi (2x)$ , where $\varphi (x) = \sum_{n=-\infty }^{\infty } \frac{1}{(x-n\pi )^{2}} - \frac{1}{\sin^{2}x}$ ), show that

\frac{1}{\sin ^{2}x} = \sum _{n=-\infty }^{\infty } \frac{1}{(x-n\pi )^{2}}.

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Problem 5075

By integration, obtain Euler's equality

\operatorname {ctg}x = \frac{1}{x}+\sum _{n \neq 0}\left(\frac{1}{x-n\pi }+\frac{1}{x+n\pi }\right) = \frac{1}{x}+\sum _{n=1}^{\infty } \frac{2x}{x^{2}-n^{2}\pi ^{2}}.

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Problem 5076

By integrating and passing from logarithms to numbers, prove the equality:

\sin x = x\prod _{n=1}^{\infty }\left(1-\frac{x^{2}}{n^{2}\pi ^{2}}\right). \qquad \text{(Euler.)}

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Problem 5077

Prove Euler's equality giving the expansion into an infinite series of fractions of the function $\operatorname {tg}x = \sum_{n=0}^{\infty } \frac{2x}{\left(\frac{2n+1}{2}\pi \right)^{2}-x^{2}}$ .

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Problem 5078

Prove Euler's equality giving the expansion into an infinite series of fractions of the function $\frac{1}{\sin x} = \frac{1}{x}+\sum_{n=1}^{\infty } \frac{(-1)^{n}\cdot 2x}{x^{2}-n^{2}\pi^{2}}$ .

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Problem 5079

Prove Euler's equality giving the expansion into an infinite series of fractions of the function $\frac{1}{\cos x} = \sum_{n=1}^{\infty } \frac{(-1)^{n}(2n-1)\pi }{x^{2}-\left(n-\frac{1}{2}\right)^{2}\pi^{2}}$ .

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Problem 5080

Prove Euler's equality giving the expansion into an infinite series of fractions of the function $\frac{1}{1-e^{-x}} - \frac{1}{x} - \frac{1}{2} = \sum_{n=1}^{\infty } \frac{2x}{x^{2}+4\pi^{2}n^{2}}$ .

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